Monday, August 10, 2009 - 15:00 , Location: Skiles 255 , Kun Zhao , School of Mathematics, Georgia Tech , Organizer:
Thursday, July 2, 2009 - 13:30 , Location: Skiles 255 , Turkay Yolcu , School of Mathematics, Georgia Tech , Organizer:
In this thesis, we extend De Giorgi's interpolation method to a class of parabolic equations which are not gradient flows but possess an entropy functional and an underlying Lagrangian. The new fact in the study is that not only the Lagrangian may depend on spatial variables, but also it does not induce a metric. Assuming the initial condition is a density function, not necessarily smooth, but solely of bounded first moments and finite entropy, we use a variational scheme to discretize the equation in time and construct approximate solutions. Moreover, De Giorgi's interpolation method reveals to be a powerful tool for proving convergence of our algorithm. Finally, we analyze uniqueness and stability of our solution in L^1.
Wednesday, July 1, 2009 - 15:30 , Location: Skiles 255 , Alan J. Michaels , School of Electrical and Computer Engineering, Georgia Tech , Organizer:
This disseratation provides the conceptual development, modeling and simulation, physical implementation and measured hardware results for a procticable digital coherent chaotic communication system.
Monday, May 11, 2009 - 13:00 , Location: Skiles 255 , Evan Borenstein , School of Mathematics, Georgia Tech , Organizer: Ernie Croot
Wednesday, April 8, 2009 - 15:00 , Location: Skiles 255 , Hwa Kil Kim , School of Mathematics, Georgia Tech , Organizer:
Tuesday, November 11, 2008 - 13:30 , Location: Skiles 269 , Stephen Young , School of Mathematics, Georgia Tech , Organizer: Annette Rohrs
Monday, November 3, 2008 - 13:30 , Location: Skiles 114 , Alex Yurchenko , School of Mathematics, Georgia Tech , Organizer:
The first part of this work deals with open dynamical systems. A natural question of how the survival probability depends upon a position of a hole was seemingly never addresses in the theory of open dynamical systems. We found that this dependency could be very essential. The main results are related to the holes with equal sizes (measure) in the phase space of strongly chaotic maps. Take in each hole a periodic point of minimal period. Then the faster escape occurs through the hole where this minimal period assumes its maximal value. The results are valid for all finite times (starting with the minimal period), which is unusual in dynamical systems theory where typically statements are asymptotic when time tends to infinity. It seems obvious that the bigger the hole is the bigger is the escape through that hole. Our results demonstrate that generally it is not true, and that specific features of the dynamics may play a role comparable to the size of the hole. In the second part we consider some classes of cellular automata called Deterministic Walks in Random Environments on \mathbb Z^1. At first we deal with the system with constant rigidity and Markovian distribution of scatterers on \mathbb Z^1. It is shown that these systems have essentially the same properties as DWRE on \mathbb Z^1 with constant rigidity and independently distributed scatterers. Lastly, we consider a system with non-constant rigidity (so called process of aging) and independent distribution of scatterers. Asymptotic laws for the dynamics of perturbations propagating in such environments with aging are obtained.